Constraints on supersymmetric grand unified theories from cosmology

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Constraints on Supersymmetric Grand Unified

Theories from Cosmology

Jonathan Rocher1†, Mairi Sakellariadou2⋆

1 Institut d’Astrophysique de Paris, GRεCO, FRE 2435-CNRS, 98bis boulevardArago, 75014 Paris, France2 Division of Astrophysics, Astronomy and Mechanics, Department of Physics,University of Athens, Panepistimiopolis, GR-15784 Zografos (Athens), Hellasand Institut d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris,France

E-mail: †[email protected], ⋆[email protected], [email protected]

Abstract. Within the context of SUSY GUTs, cosmic strings are genericallyformed at the end of hybrid inflation. However, the WMAP CMB measurementsstrongly constrain the possible cosmic strings contribution to the angular powerspectrum of anisotropies. We investigate the parameter space of SUSY hybrid (F-and D- term) inflation, to get the conditions under which theoretical predictionsare in agreement with data. The predictions of F-term inflation are in agreementwith data, only if the superpotential coupling κ is small. In particular, for SUSYSO(10), the upper bound is κ <∼ 7× 10−7. This fine tuning problem can be liftedif we employ the curvaton mechanism, in which case κ <∼ 8× 10−3; higher valuesare not allowed by the gravitino constraint. The constraint on κ is equivalent toa constraint on the SSB mass scale M , namely M <∼ 2 × 1015 GeV.The study of D-term inflation shows that the inflaton field is of the order of thePlanck scale; one should therefore consider SUGRA. We find that the cosmicstrings contribution to the CMB anisotropies is not constant, but it is stronglydependent on the gauge coupling g and on the superpotential coupling λ. Weobtain g <∼ 2 × 10−2 and λ <∼ 3 × 10−5. SUGRA corrections induce also a lower

limit for λ. Equivalently, the Fayet-Iliopoulos term ξ must satisfy√

ξ <∼ 2× 1015

GeV. This constraint holds for all allowed values of g.

PACS numbers: cos, ssb, suy

1. Introduction

High energy physics and cosmology are two complementary areas with a rich andfruitful interface. They both enter the description of the physical processes duringthe early stages of our Universe. High energy physics leads to the notion oftopological defects, which imply a number of cosmological consequences. Once wecompare the theoretical predictions of models motivated by high energy physics againstcosmological data, we induce constraints or, in other words, we fix the free parametersof the models. This is the philosophy of our study.

Even though the particle physics Standard Model (SM) has been tested to a veryhigh precision, evidence of neutrino masses [1, 2, 3] proves that one should go beyondthis model. An extension of the SM gauge group can be accomplished in the frameworkof Supersymmetry (SUSY). At present, SUSY is the only viable theory for solving the

http://arXiv.org/abs/hep-ph/0406120v2

Constraints on Supersymmetric Grand Unified Theories from Cosmology 2

gauge hierarchy problem. Moreover, in the supersymmetric standard model the gaugecoupling constants of the strong, weak and electromagnetic interactions, with SUSYbroken at the TeV-scale, meet at a single point MGUT ≃ (2−3)×1016 GeV. These arecalled Supersymmetric Grand Unified Theories (SUSY GUTs). An acceptable SUSYGUT model should be consistent with the standard model as well as with cosmology.SUSY GUTs can provide the scalar field needed for inflation, they can explain thematter-antimatter asymmetry of the Universe, and they can provide a candidate forcold dark matter, known as the lightest superparticle.

Usually models of SUSY GUTs suffer from the appearance of undesirable stabletopological defects, which are mainly monopoles, but also domain walls. Topologicaldefects appear via the Kibble mechanism [4]. A common mechanism to get rid ofthe unwanted topological defects, is to introduce one or more inflationary stages.Inflation essentially consists of a phase of accelerated expansion which took place ata very high energy scale. In addition, inflation provides a natural explanation forthe origin of the large scale structure and the associated temperature anisotropies inthe Cosmic Microwave Background (CMB) radiation. On the other hand, one hasto address the question of whether it is needed fine-tuning of the parameters of theinflationary model. This is indeed the case in some nonsupersymmetric versions ofinflation and it leads to the naturalness issue. Even though SUSY models of hybridinflation were for long believed to circumvent these fine-tuning issues, our study showsthat this may not be the case, unless we invoke the curvaton mechanism for the originof the initial density fluctuations.

The study we present here is the continuation of Ref. [5], where they wereexamined all possible Spontaneously Symmetry Breaking (SSB) schemes from a largegauge group down to the SM gauge group, in the context of SUSY GUTs. Assumingstandard supersymmetric hybrid inflation, there were found all models which areconsistent with high energy physics and cosmology. Namely, there were selectedall models which can solve the GUT monopole problem, lead to baryogenesis afterinflation and are consistent with proton lifetime measurements. That study led tothe conclusion that in all acceptable SSB patterns, the formation of cosmic strings isunavoidable, and some times it is accompanied by the formation of embedded strings.The strings which form at the end of inflation have a mass which is proportional to theinflationary scale. Here, we find the inflationary scale, which coincides with the stringmass scale. Since our analysis is within global supersymmetry, we examine whetherthe value of the inflaton is at least a few orders of magnitude smaller than the Planckscale. As we show, global supersymmetry is sufficient in the case of F-term inflation,while D-term inflation necessitates the supergravity framework.

We organise the rest of the paper as follows: In Section II, we discuss thetheoretical framework of our study. We briefly review the choice of the gauge groupswe consider and we state the results about SSB schemes allowed from particle physicsand cosmology as well as the topological defects left after the last inflationary era. Webriefly review the theory of CMB temperature fluctuations. We then discuss inflationwithin N=1 SUSY GUTs, first in the context of F-term and then in the context ofD-term inflation. In Section III, we describe our analysis as well as our findings forthe mass scale and the strings contribution to the CMB. We first discuss F-term andthen that of D-term inflation. We show that F-term inflation can be addressed inthe context of global supersymmetry. We then show that D-term inflation has tobe studied within local supersymmetry and we give the scalar potential for D-terminflation, taking into account radiative and supergravity (SUGRA) corrections. We


round up with our conclusions in Section IV. In Appendix A, we list the allowed SSBschemes of large gauge groups down to the standard model, found in Ref. [5], whichare allowed by particle physics and cosmology.

2. Theoretical Framework

2.1. Hybrid inflation within supersymmetric grand unified theories

Grand Unified Theories imply a sequence of phase transitions associated with theSSB of the GUT gauge group GGUT down to the standard model gauge groupGSM = SU(3)C × SU(2)L × U(1)Y. The energy scale at which this sequence of SSBsstarts is MGUT ∼ 3 × 1016 GeV. As our Universe has undergone this series of phasetransitions, various kinds of topological defects may have been left behind as theconsequence of SSB schemes, via the Kibble mechanism [4]. Among the various kindsof stable topological defects, monopoles and domain walls are undesirable, since theylead to catastrophic cosmological implications, while textures do not have importantcosmological consequences. To get rid of the unwanted topological defects one mayemploy the mechanism of cosmological inflation.

Considering supersymmetric grand unified theories is motivated by severalconstraints, coming from both particle physics and cosmology. It is indeed the onlyviable framework to solve the hierarchy problem. In addition, it allows for a unificationof strong, weak and electromagnetic interactions at a sufficiently high scale to becompatible with proton lifetime measurements. From the point of view of cosmology,SUSY GUTs can provide a good candidate for dark matter, namely the lightestsupersymmetric particle. Moreover, it offers various candidates for playing the role ofthe inflaton field and it can give naturally a flat direction for slow-roll inflation.

Supersymmetry can be formulated either as a global or as a local symmetry,in which case gravity is included and the theory is called supergravity. Globalsupersymmetry can be seen as a limit of supergravity and it is a good approximationprovided the Vacuum Expectation Values (VEVs) of all relevant fields are muchsmaller than the Planck mass.

In a supersymmetric theory, the tree-level potential for a scalar field is the sumof an F-term and a D-term. These two terms have rather different properties andin all proposed hybrid inflationary models only one of the two terms dominates.In supersymmetric hybrid inflation, the superpotential couples an inflaton field toa pair of Higgs fields that are responsible for one symmetry breaking in the SSBscheme. This class of inflationary models is considered natural [6] within SUSY GUTsin the sense that the only extra field which is added, except the fields needed tobuild the GUT itself, is a singlet scalar field. This extra field is however likely tobe anyway needed to build the GUT model, so that it constraints the Higgs fields toacquire a VEV. Moreover, it is not spoiled by radiative corrections and supergravitycorrections can be kept small for F-term inflation. We would like to mention thatgenerically, F-term inflation suffers from the so-called η-problem, because usually, thesupergravity corrections induce contribution of order unity to the slow roll parameterη ≡M2

PlV′′/V . This effective mass term for the inflaton field would spoil the slow roll

condition [7]. However, in the case of a minimal Kahler potential, which is what isconsidered hereafter, this problem is lifted by a cancelation of the problematic massterms. This can however be seen as a fine tunning since a minimal kahler potential isnot well motivated for example from the point of view of the string theory [7]. Finally


it has been stated that such inflationary models are successful in the sense that theydo not require any fine tuning, but we show that this last point has to be revisited.

F-term inflation can occur naturally within the GUTs framework when forexample, a GUT gauge group GGUT is broken down to the SM at an energy MGUT

according to

GGUTMGUT−−−→ H1

Minfl−−−−→Φ+Φ−

H2−→GSM , (1)

where Φ+,Φ− is a pair of GUT Higgs superfields in nontrivial complex conjugaterepresentations, which lower the rank of the group by one unit when acquiring nonzeroVEV. The inflationary phase takes place at the beginning of the symmetry breaking

H1Minfl−→ H2.F-term inflation is based on the globally supersymmetric renormalisable

superpotential

WFinfl = κS(Φ+Φ− −M2) , (2)

where S is a GUT gauge singlet left handed superfield and Φ+,Φ− are as defined above,with κ and M two constants (M has dimensions of mass) which can both be takenpositive with field redefinition. The chiral superfields S,Φ+,Φ− are taken to havecanonical kinetic terms. The above superpotential is the most general superpotentialconsistent with an R-symmetry under which W → eiβW ,Φ− → e−iβΦ− ,Φ+ →eiβΦ+, and S → eiβS. We note that an R-symmetry can ensure that the rest of therenormalisable terms are either absent or irrelevant.

D-term inflation is derived from the superpotential

WDinfl = λSΦ+Φ− , (3)

where S,Φ−,Φ+ are three chiral superfields and λ is the superpotential coupling. D-term inflation requires the existence of a nonzero Fayet-Iliopoulos term ξ, which canbe added to the lagrangian only in the presence of a U(1) gauge symmetry, underwhich, the three chiral superfields have charges QS = 0, QΦ+ = +1 and QΦ−

= −1,respectively. Thus, D-term inflation requires a scheme, like

GGUT × U(1)MGUT−−−→ H × U(1)

Minfl−−−−→Φ+Φ−

H → GSM . (4)

This extra U(1) gauge symmetry symmetry can be of a different origin [6]. In whatfollows, we consider a nonanomalous U(1) gauge symmetry. We note however thatone could instead consider a situation realised in heterotic string theories, where thereis an anomalous D-term arising from an anomalous U(1)A, which can contribute tothe vacuum energy [8]. Clearly, the symmetry breaking at the end of the inflationnaryphase implies that cosmic strings are always formed at the end of D-term hybridinflation.

In the SSB schemes studied in Ref. [5], one can naturally incorporate an eraof F-/D-term inflation. All SSB schemes from grand unified gauge groups GGUT ofrank at the most equal to 8 down to the standard model gauge group GSM × Z2

have been considered. Initially, the group GGUT was chosen to be one of thefollowing ones: SU(5), SO(10), SU(6), E6, SU(7), SO(14), SU(8), and SU(9). Inaddition, the choice of GGUT was limited to simple gauge groups which containGSM, have a complex representation, are anomaly free and take into account someof the major observationnal constraints of particle physics and cosmology. The above


Z2 symmetry is a sub-group of the U(1)B−L gauge symmetry which is contained invarious gauge groups and it plays the role of R-parity. R-parity can only appear ingrand unified gauge groups which contain U(1)B−L. There were considered as manypossible embeddings of GSM in GGUT as one can find in the literature and it wasexamined whether defects are formed during the SSB patterns of GGUT down to GSM,and of which kind they are. Assuming standard hybrid F-term inflation there weredisregarded all SSB patterns where monopoles or domain walls are formed after theend of the last possible inflationary era. In addition, there were disregarded SSBschemes with broken R-parity since the proton would decay too rapidly as comparedto Super-Kamiokande measurements. It was also required that the gauged U(1)B−L

symmetry, is broken at the end of inflation so that a non thermal leptogenesis canexplain the baryon/antibaryon asymmetry in the Universe. SU(5), SU(6), SU(7) arethus not acceptable groups for particle physics, since SU(5) leads to the formation ofstable monopoles, while minimal SU(6) and minimal SU(7) do not contain U(1)B−L.

The SSB schemes compatible with high energy physics and cosmology, as givenin detail in Ref. [5], are listed in Appendix A.

It was concluded that within the framework of the analysis of Ref. [5], there arenot any acceptable SSB schemes without cosmic strings after the last inflationary era.The analysis we present below can therefore be applied to all these models. We wouldlike to note that even if we relax the requirement that the gauged B − L symmetry isbroken at the end of inflation, the results of Ref. [5] remain unchanged.

Even if one also allows for patterns with broken R-parity at low energy, cosmicstrings formation is still very generic. To consider these cases, one should however finda mechanism to protect proton lifetime from very dangerous dimension 4 operators.Strings formed during the SSB phase transitions leading from a large gauge groupGGUT down to the GSM × Z2 are of two types: topological strings, called cosmicstrings, and embedded strings which are not topologically stable and in general theyare not dynamically stable either.

Clearly, as we discussed earlier, in the case of D-term inflation, cosmic strings arealways present at the end of the inflationary era.

2.2. Cosmic microwave background temperature anisotropies

The CMB temperature anisotropies provide a powerful test for theoretical modelsaiming at describing the early Universe. The characteristics of the CMB multipolemoments, and more precisely the position and amplitude of the acoustic peaks, aswell as the statistical properties of the CMB temperature anisotropies, can be used todiscriminate among theoretical models, as well as to constrain the parameters space.

The spherical harmonic expansion of the cosmic microwave backgroundtemperature anisotropies, as a function of angular position, is given by

δT

T(n) =

∑

ℓm

aℓmWℓYℓm(n) (5)

with

aℓm =

∫

dΩn

δT

T(n)Y ∗

ℓm(n) , (6)

where Wℓ stands for the ℓ-dependent window function of the particular experiment.


The angular power spectrum of CMB anisotropies is expressed in terms of thedimensionless coefficients Cℓ, which appear in the expansion of the angular correlationfunction in terms of the Legendre polynomials Pℓ:

⟨

0

∣

∣

∣

∣

δT

T(n)

δT

T(n′)

∣

∣

∣

∣

0

⟩

∣

∣

∣

(n·n′=cos ϑ)

=1

4π

∑

ℓ

(2ℓ+ 1)CℓPℓ(cosϑ)W2ℓ .(7)

It compares points in the sky separated by an angle ϑ. Here, the brackets denotespatial average, or expectation values if perturbations are quantised ‡. The value ofCℓ is determined by fluctuations on angular scales of order π/ℓ. The angular powerspectrum of anisotropies observed today is usually given by the power per logarithmicinterval in ℓ, plotting ℓ(ℓ+ 1)Cℓ versus ℓ. The coefficients Cℓ are related to aℓm by

Cℓ =〈∑m |aℓm|2〉

2ℓ+ 1. (8)

On large angular scales, the main contribution to the temperature anisotropies isgiven by the Sachs-Wolfe effect, implying

δT

T(n) ≃ 1

3Φ[ηlss,n(η0 − ηlss)], (9)

where Φ(η,x) is the Bardeen potential, while η0 and ηlss denote respectively theconformal times now and at the last scattering surface. Note that the previousexpression is only valid for the standard cold dark matter model.

If we assume that the initial density perturbations are due to “freezing in”of quantum fluctuations of a scalar field during an inflationary period, then thequadrupole anisotropy reads

(

δT

T

)

Q−infl

=

[

(

δT

T

)2

Q−scal

+

(

δT

T

)2

Q−tens

]1/2

, (10)

where the quadrupole anisotropy due to the scalar Sachs-Wolfe effect is(

δT

T

)

Q−scal

=1

4√

45π

V 3/2(ϕQ)

M3Pl V

′(ϕQ), (11)

and the tensor quadrupole anisotropy is(

δT

T

)

Q−tens

∼ 0.77

8π

V 1/2(ϕQ)

M2Pl

. (12)

We note that V is the potential of the inflaton field ϕ, with V ′ ≡ dV (ϕ)/dϕ, whileMPl denotes the reduced Planck mass, MPl = (8πG)−1/2 ≃ 2.43 × 1018 GeV, andϕQ is the value of the inflaton field when the comoving scale corresponding to thequadrupole anisotropy became bigger than the Hubble radius.

The number of e-foldings of inflation between the initial value of inflaton ϕi andthe final value ϕf is given by

N(ϕi → ϕf ) = −8πG

∫ ϕf

ϕi

V (ϕ)

V ′(ϕ)dϕ . (13)

The primordial fluctuations could also be generated from the quantumfluctuations of a late-decaying scalar field other than the inflaton, known as the

‡ We emphasise that Eq. (7) holds only if the initial state for cosmological perturbations of quantum-mechanical origin is the vacuum [9, 10].


curvaton field ψ [11, 12, 13, 14], whose nonvanishing initial amplitude is denotedby ψinit. During inflation the curvaton potential is very flat and ψ acquires quantumfluctuations [12]

δψinit =Hinf

2π, (14)

where Hinfl denotes the expansion rate during inflation, which is a function of theinflaton field, and it is given by the Friedmann equation

H2infl(ϕ) =

8πG

3V (ϕ) . (15)

We assumed that the effective curvaton mass is much smaller than Hinfl, sinceotherwise the quantum fluctuations of the curvaton field during inflation would bevery small and the CMB power spectrum would remain the same as in the standardadiabatic case.

At the end of the inflationary era, δψinit generates an entropy fluctuation. At latertimes, the curvaton field first dominates the energy density of the Universe, and then(during the RDE) it decays and reheats the Universe. Since the primordial fluctuationsof the curvaton field are converted to purely adiabatic density fluctuations, the effectof δψinit in the CMB angular power spectrum, which can be parametrised by themetric perturbation induced by the curvaton fluctuation, is given by [15]

Ψcurv = −4

9

δψinit

ψinit. (16)

There is no correlation between the primordial fluctuations of the inflaton and curvatonfields.

As we have explicitely shown in Ref. [5], the end of the inflationary era isaccompanied by strings formation §, which are cosmic strings (topological defects),sometimes accompagnied by embedded strings (not topologically stable and in generalnot dynamically stable either). Let us first calculate the contribution to the quadrupoletemperature anisotropy coming from the cosmic strings network.

At this point, we would like to bring to the attention of the reader that sincethe calculation of the angular power spectrum induced by a cosmic strings networkrelies on heavy numerical simulations, this issue remains still open. More precisely, toobtain the power spectrum from numerical simulations with cosmic strings, one musttake into account the “three-scale model” [20] of cosmic strings networks, the small-scale structure (wiggliness) of the strings, the microphysics of the strings network,as well as the expansion of the Universe. This is indeed a rather difficult task, andto our knowledge no currently available simulation leading to the Cstrings

ℓ includesall of them. Moreover, all simulations of cosmic strings are referred to Nambu-Gotostrings, and this is probably the most unrealistic case ‖ At least, but not last, there

§ Some authors have proposed mechanisms which could avoid cosmic strings production at theend of hybrid inflation. This is for example realised by adding a nonrenormalisable term in thesuperpotential [16], or by adding an additional discrete symmetry [17], or by considering GUTsmodels based on nonsimple and more complicated groups [18]. Moreover, by introducing a new pairof charged superfields in the framework of an N=2 supersting version of D-term inflation the stringswhich form are nontopological [19]. This last model was shown [19] to satisfy the CMB constraints.‖ The nature of cosmic strings formed in the context of our models within SUSY GUTs and theircosmological role is studied in Ref. [21]. One has to examine whether the fermion zero modes whichare intrinsic in supersymmetric models of cosmic strings lead to the production of vortons, whichmay result to a cosmological problem [22]. The microphysics of cosmic string solutions to N=1supersymmetric abelian Higgs models has been studied in Ref. [23].


is the issue of the appropriate initial conditions for the time evolution of the cosmicstrings network. More precisely, we do not know of any numerical simulation leadingto the Cstrings

ℓ , where the initial configuration of the strings network was other thanthe one obtained by assigning at random values to a phase variable on a cubic lattice.One should probably study whether the strings network is in the low or high densityregime ¶.

Nevertheless, in what follows we use recent results [26] based on Nambu-Gotolocal strings simulations in a Friedmann–Lemaıtre–Roberston–Walker spacetime andwe assume

(

δT

T

)

cs

∼ (9 − 10)Gµ with µ = 2π〈χ〉2 , (17)

where 〈χ〉 is the Vacuum Expectation Value (VEV) of the Higgs field responsible forthe formation of cosmic strings.

2.3. Inflation in supersymmetric grand unified theories

In what follows, we first discuss inflation where the F-term dominates (F-terminflation) and then we address inflation where the D-term dominates (D-terminflation).

2.3.1. F-term inflation F-term inflation is based on the globally supersymmetricrenormalisable superpotential Eq. (2). The scalar potential V can be obtained fromEq. (2) and it reads

V (φ+, φ−, S) = |FΦ+ |2 + |FΦ−|2 + |FS |2 +

1

2

∑

a

g2aD

2a , (18)

where the F-term is such that+ FΦi≡ |∂W/∂Φi|θ=0, with Φi = Φ+,Φ−, S, and

Da = φi (Ta)ij φ

j + ξa , (19)

with a the label of the gauge group generators Ta, ga the gauge coupling, and ξa theFayet-Iliopoulos term. By definition, in the F-term inflation the real constant ξa iszero; it can only be nonzero if Ta generates a U(1) group.

In the case of F-term inflation, the potential V as a function of the complex scalarcomponent of the respective chiral superfields Φ+,Φ−, S reads

V F(φ+, φ−, S) = κ2|M2 − φ−φ+|2 + κ2|S|2(|φ−|2 + |φ+|2) + D − terms. (20)

Assuming that the F-terms give rise to the inflationary potential energy density, whilethe D-terms are flat along the inflationary trajectory, one may neglect them duringinflation. The D-terms may play an important role in determining the trajectory andin stabilising the noninflaton fields.

The potential has two minima: one valley of local minima (V = κ2M4 ≡ V0),for S greater than its critical value Sc = M with φ+ = φ− = 0, and one globalsupersymmetric minimum (V = 0) at S = 0 and φ+ = φ− = M . Imposing chaotic

¶ When the energy density of the strings network is low, the dominant part of the strings is in theform of closed loops of the smallest allowed size. At a certain critical density the strings networkundergoes a phase transition characterised by the formation of infinite strings [24, 25].+ The notation |∂W/∂Φi|θ=0 means that one has to take the scalar component (with θ = θ = 0 inthe superspace) of the superfields once one differentiates with respect to the superfields Φi. This isthe reason for which the potential V is a function of the scalar fields φi.


initial conditions, i.e. S ≫ Sc, the fields quickly settle down the valley of local minima.In the slow roll inflationary valley (φ+ = φ− = 0, |S| ≫ M), the ground state of thescalar potential is different from zero, meaning that SUSY is broken. In the tree level,along the inflationary valley the potential V = V0 is constant and thus it is perfectlyflat. A slope along the potential can be generated by including the one-loop radiativecorrections which are small as compared to V0. Thus, the scalar potential acquires alittle tilt which helps the scalar field S to slowly roll down the valley of minima. TheSSB of SUSY along the inflationary valley by the vacuum energy density κ2M4 leads toa mass splitting in the superfields Φ+,Φ−. One gets [28] a Dirac fermion with a masssquared term κ2|S|2 and two complex scalars with mass squared terms κ2|S|2±κ2M2.This implies that there are one-loop radiative corrections to V along the inflationaryvalley, which can be calculated using∗ the Coleman-Weinberg expression [27]

∆V1−loop =1

64π2

∑

i

(−1)Fim4i ln

m2i

Λ2, (21)

where the sum extends over all helicity states i, with fermion number Fi and masssquared m2

i ; Λ stands for a renormalisation scale. Thus, the effective potentialreads [28, 29, 30, 31, 32]V F

eff(|S|) = V0 + [∆V (|S|)]1−loop

= κ2M4

1+κ2N32π2

[

2 ln|S|2κ2

Λ2+(z+1)2 ln(1+z−1)+(z−1)2 ln(1−z−1)

]

, (22)

where

z =|S|2M2

≡ x2 , (23)

and N stands for the dimensionality of the representations to which the fields φ+, φ−belong.

Employing the above found expression, Eq. (22), for the effective potential wecalculate in the next section the inflaton and cosmic strings contributions to the CMBtemperature anisotropies.

2.3.2. D-term inflation In the context of global supersymmetry, D-term inflationis derived from the superpotential Eq. (3). In the global supersymmetric limit,Eqs.(18), (3) lead to the following expression for the scalar potential

V D(φ+, φ−, S) = λ2[

|S|2(|φ+|2 + |φ−|2) + |φ+φ−|2]

+g2

2(|φ+|2−|φ−|2+ξ)2 , (24)

where g is the gauge coupling of the U(1) symmetry and ξ is a Fayet-Iliopoulos term,chosen to be positive.

The potential has two minima. There is one global minimum at zero, reached for|S| = |φ+| = 0 and |φ−| =

√ξ. There is also one local minimum, found by minimising

the potential for fixed values of S with respect to the other fields. This local minimumis equal to V0 = g2ξ2/2, reached for |φ+| = |φ−| = 0, with |S| > Sc ≡ g

√ξ/λ. As in

the previous discussed case (F-term inflation), also here the SSB of supersymmetry inthe inflationary valley introduces a splitting in the masses of the components of the

∗ This expression has been derived in the case of a Minkowski background. However, during inflationthe background geometry is given by the De Sitter metric, and therefore, strictly speaking, one shouldnot use the standard Coleman-Weinberg expression, but one should instead find the correspondingexpression in a De Sitter background.


chiral superfields Φ+ and Φ−. As a result, we obtain two scalars with squared massesm2

± = λ2|S|2 ± g2ξ and a Dirac fermion with squared mass λ2|S|2.For arbitrary large S the tree level value of the potential remains constant and

equal to V0 = (g2/2)ξ2, thus S plays naturally the role of an inflaton field. Assumingchaotic initial conditions |S| ≫ Sc one can see the onset of inflation. Along theinflationary trajectory the F-term vanishes and the Universe is dominated by the D-term, which splits the masses in the Φ+ and Φ− superfields, resulting to the one-loopeffective potential for the inflaton field.

The radiative corrections given by the Coleman-Weinberg expression Eq. (21)lead to the following effective potential for D-term inflation

V Deff(|S|) = V0 + [∆V (|S|)]1−loop

=g2ξ2

2

1+g2

16π2

[

2 ln|S|2λ2

Λ2+(z+1)2 ln(1+z−1)+(z−1)2 ln(1−z−1)

]

, (25)

with

z =λ2|S|2g2ξ

≡ x2 . (26)

Employing the above found expression for the effective potential, Eq. (25), wecalculate in the next section the inflaton and cosmic strings contributions to the CMBtemperature anisotropies.

3. Energy scale of inflation and inflaton/cosmic strings contributions tothe CMB

3.1. F-term inflation in global supersymmetry

Assuming V ≃ κ2M4, while using the exact expression for the potential as given inEq. (22) for calculating V ′ ≡ dV/dS, we obtain

V ′(|S|) =2a

|S|z f(z) , (27)

with

a =κ4M4N

16π2, (28)

and

f(z) = (z + 1) ln(1 + z−1) + (z − 1) ln(1 − z−1) . (29)

Equation (11) implies [31, 32](

δT

T

)

Q−scal

∼ 1√45

√

NQ

NM2

M2Pl

x−1Q y−1

Q f−1(x2Q) , (30)

and Eq. (13) leads to

NQ =4π2

κ2NM2

M2Pl

y2Q , (31)

with

y2Q =

∫ x2Q

1

dz

zf(z), (32)


and

xQ =|SQ|M

. (33)

We remind to the reader that the index Q denotes the scale responsible for thequadrupole anisotropy in the CMB.

The coupling κ is related to the mass scale M , through the relation

M

MPl=

√

NQN κ

2 π yQ. (34)

Equation (12) implies [31, 32](

δT

T

)

Q−tens

∼ 0.77

8πκM2

M2Pl

. (35)

Employing Eq. (17) in the case of F-term inflation, we express the cosmic stringscontribution as

(

δT

T

)

cs

∼ 9

4

(

M

MPl

)2

. (36)

Therefore, the total quadrupole anisotropy[

(

δT

T

)

Q−tot

]2

=

[(

δT

T

)

scal

]2

+

[(

δT

T

)

tens

]2

+

[(

δT

T

)

cs

]2

(37)

is explicitely given by(

δT

T

)

Q−tot

∼

y−4Q

(

κ2N NQ

32π2

)2[64NQ

45N x−2Q y−2

Q f−2(x2Q)+

(

0.77κ

π

)2

+324]1/2

, (38)

where the (δT/T )totQ is normalised to the Cosmic Background Explore (COBE)

data [33], namely (δT/T )COBEQ ∼ 6.3× 10−6. For given values of κ,NQ,N , the above

equation can be solved numerically for xQ, and then employing Eqs. (32) and (34), oneobtains yQ and M . We assume NQ = 60 and we find the inflationary scale M whichis proportional to the string mass scale, as a function of the superpotential couplingκ for three values of N . We choose N = 27,126,351, which correspond to realisticSSB schemes in SO(10) or E6 models. The results are shown in Fig. 1 below.

Clearly, the mass scale is of the order of 1015 GeV, and it grows very slowly withN . Since it will be useful later to know approximately the evolution of the massparameter M with respect to the coupling κ, we fit the curve for N = 126, which isshown in Fig. 1, by

M(κ) ∼

4.1 × 1015 + 5.3 × 1012 ln(κ) GeV if κ ∈ [2 × 10−5, 10−2] ,1.1 × 1016 + 6.3 × 1014 ln(κ) GeV if κ ∈ [10−7, 2 × 10−5] .

(39)

The dimensionless coupling κ is subject to the gravitino constraint, as well as tothe constraint imposed by the CMB temperature anisotropies. As we show below, thestrongest constraint on κ is imposed by the measured CMB temperature anisotropies;the cosmic strings contribution to the power spectrum should be strongly suppressedin order not to contradict the data.

Firstly, we calculate the upper bound on the dimensionless superpotentialcoupling κ, as imposed by the gravitino constraint. After the inflationary era, ourUniverse enters the high entropy radiation dominated phase via the reheating process,


1.´10-7 1.´10-6 0.00001 0.0001 0.001 0.01Κ

1

1.5

2

2.5

3

3.5

4

M H´1015 GeVL

Figure 1. Evolution of the inflationary scale M in units of 1015 GeV as a functionof the dimensionless coupling κ. The three curves correspond to N = 27 (curvewith broken line), N = 126 (full line) and N = 351 (curve with lines and dots).

during which the inflaton energy density decays perturbatively into normal particles.This process is characterised by the reheating temperature TRH. In order to have asuccessful reheating it is important not to create too many gravitinos.

Within the Minimal Supersymmetric Standard Model (MSSM), and assuming asee-saw mechanism to give rise to massive neutrinos, the reheating temperature is [35]

TRH ≈ (8π)1/4

7(ΓMPl)

1/2 , (40)

with Γ the decay width of the oscillating inflaton and the Higgs fields into right-handedneutrinos [35],

Γ =1

8π

(

Mi

M

)2

minfl , (41)

minfl the inflaton mass, and Mi the right handed neutrino mass eigenvalues. Equations(34), (40), (41) lead to

TRH ∼ 1

12

(

60

NQ

)1/4 (1

N

)1/4

y1/2Q Mi . (42)

The reheating temperature must satisfy the gravitino constraint TRH ≤ 109 GeV [36].This constraint implies strong bounds on the Mi’s, which satisfy the inequalityMi < minfl/2, where the inflaton mass is

minfl =√

2κM . (43)

The strong bounds on Mi lead to quite small corresponding dimensionless couplingsγi. The two heaviest neutrino are expected to have masses of the order of M3 ≃ 1015

GeV and M2 ≃ 2.5× 1012 GeV respectively [34]. This implies that for yQ of the orderof 1, Mi in Eq. (40) cannot be identified with the heaviest or the next to heaviestright handed neutrino, otherwise the reheating temperature would be higher than theupper limit imposed by the gravitino bound. Thus, Mi is M1, with M1 ∼ 6 × 109

GeV [34]. (This value is in agreement with the mass suggested in Ref. [35] for themass of the right handed neutrino into which the inflaton disintegrates.)


Therefore, using Eqs. (40), (41) and (43), the gravitino constraint on the reheatingtemperature implies,

√2

14π1/4M1

[

MPl

M(κ)

]1/2 √κ ≤ 109GeV . (44)

Using the fit of the function M(κ) given in Eq. (39), it is possible to evaluatenumerically

√

κ/M(κ) and find an upper limit on the allowed values for the couplingof the superpotential,

κ <∼ 8 × 10−3 , (45)

in agreement with the upper bound found in Ref. [37].Secondly, we proceed with a strongest constraint on κ, imposed from the

measurements on the CMB temperature anisotropies. The cosmic strings contributionto the quadrupole can be calculated from

Acs ∼

(

δTT

)cs

Q(

δTT

)COBE

Q

2

, (46)

and it is shown in Fig. 2.

1.´10-7 1.´10-6 0.00001 0.0001 0.001 0.01Κ

0.1

0.51

510

50100

Cosmic strings contribution H%L

Figure 2. Evolution of the cosmic strings contribution to the quadrupoleanisotropy as a function of the coupling of the superpotential, κ. The threecurves correspond to N = 27 (curve with broken line), N = 126 (full line) andN = 351 (curve with lines and dots).

As one can see, for small values of the coupling κ the cosmic strings contributiondepends on the value of N . Thus, we should find the value of N required for theallowed SSB patterns leading to GSM × Z2.Within the framework of our study, the SSB schemes allowed from particle physicsand cosmology are explicitely found in Ref. [5] and we recall them in Appendix A. Theparameter N is the dimensionality of the Higgs fields representation that generate theSSB; these fields are coupled to the inflaton. As explained in Ref. [5], the inflationaryera should take place after the last formation of monopoles and/or domain walls sincethese objects are incompatible with cosmology. Therefore, there are just very fewchoices for the SSB stage where inflation can be placed. The value of the parameterN depends on the GUT gauge group and the SSB scheme where inflation takes place.


For SO(10), under the requirement that R-parity is conserved down to low energies,N = 126. For E6, the Higgs representations can be N = 27 or N = 351. Our resultsdepend slightly on the choice of N and in what follows, we focus mainly on N = 126.

Comparing the results we obtained for the cosmic strings contribution to theCMB fluctuations as a function of the superpotential coupling κ, (see, Fig. 2) againstthe cosmic strings contribution allowed from the measurements, we can constrain κ.Already BOOMERanG [38], MAXIMA [39] and DASI [40] experiments imposed [41]an upper limit on Acs, which is Acs <∼ 18%. Clearly, the limit set by the WilkinsonMicrowave Anisotropy Probe (WMAP) measurements [42] should be stronger. Arecent Bayesian analysis in a three dimensional parameter space [43] have shown thata cosmic strings contribution to the primordial fluctuations Acs higher than 9% isexcluded up to 99% confidence level.

We state below the constraint we found for the dimensionless parameter κ, as afunction of N ,

κ <∼ 7 × 10−7 × 126

N , (47)

which holds for N ∈ 1,16,27,126,351. This constraint on κ, Eq. (47), is inagreement with the one found in Ref. [44].

Both, the CMB lower bound and the gravitino upper bound on κ are summarisedin Fig. 3.

1.´10-7 1.´10-6 0.00001 0.0001 0.001 0.01Κ

1

2

5

10

20

50

100


CMB

κ

Gravitino

Figure 3. Constraints on the single parameter κ of the model. The gravitinoconstraint implies κ ≤ 8 × 10−3. The allowed cosmic strings contribution to theCMB angular power spectrum implies κ <∼ 7 × 10−7, for N = 126.

The implications of this constraint on the single free parameter of the F-terminflationary model are important. Hybrid F-term inflation was known to havean appealing feature as compared to the most elegant inflationary scenario withinnonsupersymmetric theories, i.e. chaotic inflation. Namely, it was believed that therewas no need of fine tuning which will set a very small value for the coupling of thesuperpotential. This nice feature has been disappeared once we compare theoreticalpredictions against cosmological data. As we show below, one possible way out isto employ the curvaton mechanism. Another implication of this constraint, quite


important for cosmology, is the fact that this constraint on κ can be converted into aconstraint on the mass parameter M . We would like to remind to the reader that thisparameter controls the mass of the strings formed, as well as the inflationary scale.Using the CMB limit on the cosmic strings contribution, we can also obtain that, upto 99% of confidence level,

M <∼ 2 × 1015 GeV . (48)

As we show below, this constraint is robust since it holds even when the curvatonmechanism is involved, and for all values of N .

In the curvaton scenario, the curvaton field is responsible for the generation of theprimordial fluctuations. The curvaton is a scalar field, that is subdominant during theinflationary era as well as at the beginning of the radiation dominated era which followsthe inflationary phase. The effective curvaton mass is assumed to be much smallerthat the Hubble parameter during the inflationary phase. Within the framework ofsupersymmetry, which is the context of our study, one expects the existence of scalarfields. Thus, it is reasonable to expect that one of them could indeed play the roleof the curvaton field. In addition, in the class of models we are considering, one mayexpect that it may exist a natural candidate for the curvaton field. As we have shownin Ref. [5], in many acceptable SSB schemes apart from the formation of topologicalcosmic strings we have the formation of embedded strings, which are topologicallyand dynamically unstable. If the decay product of embedded strings can give a scalarfield before the onset of inflation, then such a scalar field could play the role of thecurvaton field.

Assuming the existence of a curvaton field, there is an additional contribution tothe temperature anisotropies. Thus,

[(

δT

T

)

tot

]2

=

[(

δT

T

)

infl

]2

+

[(

δT

T

)

cs

]2

+

[(

δT

T

)

curv

]2

. (49)

The inflaton contribution has a scalar and a tensor part, however since the tensor partin the supersymmetric hybrid inflation is always much smaller than the scalar one, weneglect it.

Since the primordial curvaton fluctuation is converted to purely adiabatic densityfluctuations, the curvaton contribution in terms of the metric perturbation reads

(

δT

T

)

curv

≡ Ψcurv

3, (50)

and from Eq. (16) we have(

δT

T

)

curv

= − 4

27

δψinit

ψinit. (51)

For V ≃ κ2M4, Eqs. (14), (15), (34) and (51) imply[(

δT

T

)

curv

]2

= y−4Q

(

κ2NNQ

32π2

)2 [(16

81π√

3

)

κ

(

MPl

ψinit

)]2

, (52)

We solve Eq. (49) using Eq. (52) and we obtain the different contributions of thethree sources of anisotropies: inflaton, cosmic strings and curvaton. Their respectivecontributions as a function of κ, or ψinit, are shown in Fig. 4. for a fixed value ofκ, the cosmic strings contribution decreases as ψinit decreases, while the curvatoncontribution becomes dominant. It is thus possible to use the WMAP constraint to


1013 1012 1011 5´1010 1010Ψ init HGeVL

20

40

60

80

100

Contribution H%L Κ=10-5, N=126

3´10-3 10-3 10-4 10-5 10-6Κ

20

40

60

80

100

Contribution H%L Ψ init=1013 GeV, N=126

Figure 4. The cosmic strings (dark gray), curvaton (light gray) and inflaton(gray) contributions to the CMB temperature anisotropies as a function of thethe initial value of the curvaton field ψinit, and the superpotential coupling κ, forN = 126.

limit ψinit. The upper bound on ψinit depends on the coupling κ. In addition, one hasto impose the gravitino constraint. Therefore, a coupling bigger than 10−2 is excluded.More precisely,

ψinit <∼ 5 × 1013( κ

10−2

)

GeV . (53)

The above constraint is valid only if κ is in the range [10−6, 1], while for smallervalues of κ, the cosmic strings contribution is smaller than the WMAP limit for anyvalue of the curvaton field (see, Fig. 4).

3.2. F-term inflation and supergravity corrections

We proceed with the supergravity corrections. The scalar potential in supergravityhas the general form [45]

V =eG

M4Pl

[

Gi(G−1)i

jGj − 3

]

with G =K

M2Pl

+ ln|W |2M6

Pl

, (54)

where the Kahler potential K(φ, φ∗) is a real function of the complex scalar fieldsφi, and their Hermitian conjugates φi

∗. The φi are scalar components of the chiral


superfields Φi. We adopted the following notations

Gi ≡ ∂G

∂φi, Gj ≡ ∂G

∂φj∗. (55)

Assuming that the D-term is flat along the inflationary trajectory, one can ignoreit during F-term inflation. Thus, we only consider the F-term in the above equation.If we expand K =

∑

i |φi|2+ · · ·, the corrections we get to the lowest order inflationarypotential lead to the scalar potential

V =∑

i

∣

∣

∣

∣

∂W

∂φi

∣

∣

∣

∣

2(

1 +1

M2Pl

∑

i

|φi|2 + · · ·)

+ other terms . (56)

As it was shown in Ref. [7], assuming the minimal form for the Kahler potential Kand minimising the scalar potential V with respect to φ− and φ+ for |S| > M , weobtain that the SUGRA correction to the scalar potential is

VSUGRA = κ2M4

[

1

8

|S|4M4

Pl

+ · · ·]

. (57)

The effective scalar potential becomes

Veff = κ2M4

1 +κ2N32π2

[

2 ln|S|2κ2

Λ2+ (z + 1)2 ln(1 + z−1) + (z − 1)2 ln(1 − z−1)

]

+1

8

|S|4M4

Pl

. (58)

In the literature (e.g., see Ref. [28]), some authors consider only two terms for thescalar potential. Namely, they consider only the first term of the radiative correctionsand the term coming from the SUGRA correction. However, this assumption holdsonly if xQ ≫ 1. In our study we have found that the order of magnitude for xQ isO(1) except for high values of the superpotential coupling κ (namely κ ≥ 10−2), butthose are forbidden by the gravitino constraint. In conclusion, we find that all termscoming from the radiative corrections have to be taken into account.

Since |SQ|/MPl <∼ 10−3, the potential and its first derivative is not significantlyaffected by the SUGRA corrections. Thus, including these corrections we do notexpect to find any difference in the δT/T . This is indeed what one expects, since thevalue of the inflaton field, in the absence of SUGRA corrections, stays far below thePlanck mass.

In conclusion, one can indeed neglect the supergravity corrections in theframework of F-term inflation.

3.3. D-term inflation in global supersymmetry

In what follows, we derive in the framework of global SUSY the cosmic stringscontribution to the temperature anisotropies of the CMB. We use V ≃ V0 = g2ξ2/2,while we employ Eq. (25) to derive the exact expression for V ′ ≡ ∂V (|S|)/∂|S|, whichreads

V ′(|S|) =2b

|S|z f(z) , (59)

where f(z) is the same as in the F-term, Eq. (29), and

b =g4ξ2

16π2. (60)


The next step is to express the number of e-foldings NQ. Employing the expressionswritten above for V and V ′ and using Eq. (13), we obtain

NQ =2π2

λ2

ξ

M2Pl

∫ zQ

zend

dz

zf(z), (61)

with

zend =λ2|Send|2g2ξ

. (62)

One has to evaluate the value of the inflaton field at the end of inflation. Theinflationary era ends when one of the two following conditions is reached:

• In the global minimum of the potential, 〈φ+〉 or 〈φ+〉 is not zero anymore. Thismeans that one of the two fields acquires a positive effective mass squared. Letus call zSB the value of z for which this condition is realized.

• The slow roll conditions cease to be satisfied. Within supersymmetric hybridinflation, the slow roll parameter ǫ is generically very small, and the slow rollcondition becomes just η ∼ 1. We denote by zSR the value of z for which the slowroll condition ceases to be satisfied.

Thus,

zend = max(zSB, zSR) . (63)

Let us calculate the values of zSB and zSR. Firstly, zSB can be read from the quadraticpart (for the fields φ+ and φ−) of the potential as it is given by Eq. (24). Namely,from the Lagrangian

L = λ2|S|2(|φ+|2 + |φ−|2) + g2ξ(|φ+|2 − |φ−|2) , (64)

we get zSB = 1.Secondly, the end of the slow roll phase of inflation is reached when the slow rollparameter η becomes of the order of unity,

η ≡M2Pl

[

V ′′(S)

V (S)

]

∼ 1 . (65)

For our model,

η =λ2

4π2

(

M2Pl

ξ

)

g(z) , (66)

where

g(z) = (3z + 1) ln(1 + z−1) + (3z − 1) ln(1 − z−1) . (67)

Equation (65) can be solved provided λ <∼ 4×10−3. This is a reasonable conditionsince higher values of λ would be forbidden by the gravitino constraint (see discussionin III.A). For λ <∼ 4 × 10−3, ξ/λ2 >∼ 1036 and as one can see from the shape of thefunction g(z), the two solutions of Eq. (65) are very close to 1. This implies zSR ∼ 1,and the number of e-foldings reads

NQ =2π2

λ2

ξ

M2Pl

y2Q , (68)

where yQ is given by Eq. (32) with xQ = λ|SQ|/(g√ξ).


To measure the weight of the cosmic strings contribution, we suppose three sourcesfor the quadrupole anisotropy of the CMB, namely

[(

δT

T

)

tot

]2

=

[(

δT

T

)

scal

]2

+

[(

δT

T

)

tens

]2

+

[(

δT

T

)

cs

]2

, (69)

and we calculate each term using Eqs. (11), (12), (17) respectively. In this lastequation, the VEV of the Higgs field responsible of the cosmic strings formation is√ξ.

Thus, eliminating ξ with Eq. (68), we obtain(

δT

T

)

Q−tot

∼

y−4Q

(

λ2NQ

16π2

)2[16NQ

45x−2

Q y−2Q f−2(x2

Q)

+

(

0.77g√2π

)2

+ 324]1/2

, (70)

where (δT/T )totQ is normalised to the COBE data, i.e., (δT/T )COBEQ ∼ 6.3×10−6. For

given values of λ and NQ we solve numerically Eq. (70) to get xQ.Regarding D-term inflation we find that, as in the F-term case, the tensor

contribution to the δT/T is negligible. For λ <∼ 4 × 10−3, one can compute the massscale MD =

√ξ as a function of λ. We find that ξ, which can be seen as a mass

parameter, has a very similar shape as the one found for M in the case of F-terminflation (see, Fig. 1). Moreover, we obtain that the contribution of the cosmic stringsto the quadrupole anisotropy is very similar to the F-term case. In addition, to beconsistent with the CMB data, λ in D-term inflation should obey a similar limit tothe one found for κ in F-term inflation, namely λ <∼ 3 × 10−5. These results aresummarised in Fig. 5.

At this point we would like to bring to the attention of the reader that our findingsregarding D-term inflation in SUSY, disagree with some results stated in previousstudies♯ (see eg., Ref. [6]). The reason is that in our analysis, we take into accountall terms of the radiative corrections and, as we have already shown in our discussionon F-term inflation, these terms have an essential impact on the determination of thecosmic strings contribution to the CMB.

Knowing xQ, one can compute from Eq. (26) the order of magnitude of thecorresponding inflaton field SQ. Even though Eq. (70) is very similar to the one wehad in the case of F-term hybrid inflation, i.e., Eq. (38), the conclusions for the valueof the inflaton field as compared to the Planck mass are quite different. This is dueto the dependence of xQ on the coupling λ, which makes the inflaton field SQ to beof the order of the Planck mass or higher.

The analysis we presented above implies that global supersymmetry is insufficientto study D-term inflation. It is clear that only in the context of supergravity one cantreat correctly the issue of D-term inflation and its cosmological implications.

3.4. D-term inflation in supergravity

The case of D-term inflation has to be addressed within the framework of supergravity,since as we have shown in the previous subsection, the fields are not negligible ascompared to the Planck mass.

♯ More precisely, we disagree with the statement that in the context of D-term inflation cosmic

strings contribute to the Cℓ’s up to the level of 75%.


1.´10-6 0.00001 0.0001 0.001Λ

1

1.5

2

2.5

3

3.5

4

"###Ξ H´1015 GeVL

1.´10-6 0.00001 0.0001 0.001Λ0.1

0.51

510

50100


Figure 5. On the left, evolution of the mass scale√

ξ as a function of the couplingλ. On the right, evolution of the cosmic strings contribution to the quadrupoleanisotropy as a function of the coupling of the superpotential, λ. These plots arederived in the framework of SUSY.

Inflation is still derived from the superpotential

WDinfl = λSΦ+Φ− . (71)

The F-term part of the scalar potential is given by Eq. (54). Considering also theD-term, the total scalar potential reads

V =eG

M4Pl

[

Gi(G−1)i

jGj − 3

]

+1

2[Ref(Φi)]

−1∑

a

g2aD

2a , (72)

where f(Φi) is the gauge kinetic function and

G =K

M2Pl

+ ln|W |2M6

Pl

. (73)

The Kahler potential K(φi, φ∗i ) is a real function of the complex scalar fields φi,

and their Hermitian conjugates φi∗ where the φi are scalar components of the chiral

superfields Φi. Upper (lower) indices (i, j) denote derivatives with respect to φi (φi∗).For the D-term part, ξa is the Fayet-Iliopoulos term, ga the coupling of the U(1)a

symmetry, which is generated by Ta and under which the chiral superfields S, Φ+ andΦ− have charges 0, +1 and −1 respectively. Finally,

Da = φi(Ta)ijK

j + ξa . (74)


In what follows, we assume†† the minimal structure for f(Φi) (i.e., f(Φi)=1) and takethe minimal Kahler potential given by

K = |φ−|2 + |φ+|2 + |S|2 . (75)

Therefore, as it was found in [47, 48], the scalar potential reads

V DSUGRA = λ2 exp

( |φ−|2 + |φ+|2 + |S|2M2

Pl

)[

|φ+φ−|2(

1 +|S|4M4

Pl

)

+ |φ+S|2(

1 +|φ−|4M4

Pl

)

+ |φ−S|2(

1 +|φ+|4M4

Pl

)

+ 3|φ−φ+S|2M2

Pl

]

+g2

2

(

|φ+|2 − |φ−|2 + ξ)2

. (76)

As in the case of global supersymmetry, for S > Sc the potential is minimisedfor |φ+| = |φ−| = 0 and therefore, at the tree level, the potential in the inflationaryvalley is constant, V0 = g2ξ2/2. However, despite what is sometines claimed, thisdoes not mean that the effective inflationary potential is identical to the one found inthe case of global supersymmetry. One has to compute the radiative corrections, andmust first calculate the effective masses of the components of the superfields Φ+ andΦ−. Extracting the quadratic terms from the potential, given by Eq. (76), the scalarcomponents φ+ and φ− get squared masses

m2± = λ2|S|2 exp

( |S|2M2

Pl

)

± g2ξ . (77)

To calculate the masses of the fermionic components Ψ+ and Ψ−, one has to use theLagrangian of the fermionic sector,

Lfermion =1

2eG/2

[

−Gij −GiGj +Gijk (G−1)k

l Gl]

ΨiLΨjR +h.c. , (78)

where ΨiL and ΨjR are left and right components of the Majorana spinors Ψ+ andΨ−. Using Eq. (75), one gets a Dirac fermion with squared mass

m2f = λ2|S|2 exp

( |S|2M2

Pl

)

. (79)

We calculate the radiative corrections [∆V (|S|)]1−loop using the Coleman-Weinbergexpression, Eq. (21). Thus, we compute the full effective scalar potential during D-term inflation, within the context of SUGRA

V D−SUGRAeff = V0 + [∆V (|S|)]1−loop

=g2ξ2

2

1 +g2

16π2

[

2 lnλ2|S|2

Λ2exp

( |S|2M2

Pl

)

+ (z + 1)2 ln(1 + z−1) + (z − 1)2 ln(1 − z−1)

]

(80)

with

z =λ2|S|2g2ξ

exp

( |S|2M2

Pl

)

≡ x2 . (81)

††This is the simplest supergravity model and in general the Kahler potential can be a morecomplicated function of the superfields. This is generally the case for supergravity theories derivedfrom superstring theories.


The above potential is a generalisation of the SUSY D-term potential given byEq. (25). The end of inflation is achieved when one of the two conditions is satisfied:(i) the symmetry is spontaneously broken, or (ii) the slow roll condition is violated.By definition of the variable z, the symmetry breaking condition is still

zSB = 1 . (82)

For the slow roll condition, one has to compute the slow roll parameter η and then tosolve η ∼ 1. With the effective potential given by Eq. (80), one can compute its firstderivative

V ′(|S|) =2b

|S| zf(z)

(

1 +|S|2M2

Pl

)

. (83)

Assuming V ≃ V0 and using the exact expression for V ′, the η parameter reads

η(z) =

(

g2

16π2

M2Pl

|S|2)

z

[

g(z) +|S|2M2

Pl

h3(z) +|S|4M4

Pl

h4(z)

]

, (84)

where g(z) has been introduced for D-term inflation in Eq. (67), and h3(z), h4(z) aregiven by

h3(z) = (9z + 5) ln(1 + z−1) + (9z − 5) ln(1 − z−1) , (85)

h4(z) = (4z + 2) ln(1 + z−1) + (4z − 2) ln(1 − z−1) . (86)

Supergravity corrections become essential for S at least one order of magnitude bigger

that MPl, and under this condition the solutions of η(z) = 1 are z(1)SR = 1+ and

z(2)SR = 1−. Thus, z at the end of inflation, as defined by Eq. (63), is zend = 1.

The more complicated expression for z(|S|) makes the calculation of the differentcontributions to the CMB power spectrum trickier. The cosmic strings contributionremains the same as the one computed in the case of SUSY D-term inflation, sinceit depends only on the scale

√ξ. Concerning the inflaton contribution, we can use

the dominant term of the potential V ≃ V0 and the exact expression for the firstderivative, Eq. (83), to write the number of e-foldings

NQ =2π2

g2yQ(xQ, λ, ξ, g) , (87)

where

yQ(xQ, λ, ξ, g) =

∫ x2Q

1

W(z(g2ξ)/(λ2M2Pl))

z2f(z)[1 + W(z(g2ξ)/(λ2M2Pl))]

2dz . (88)

We note that in the above definition of yQ, the function W(x) is the “W-Lambertfunction”, i.e., the inverse of the function F (x) = xex. Setting c ≡ (g2ξ)/(λ2M2

Pl), thenumber of e-foldings NQ becomes a function of only c and xQ, once g is fixed, and itis shown in Fig. 6. Imposing NQ = 60 allows us to find a numerical relation betweenc and xQ, as illustrated in Fig. 6.

Using this relation between c and xQ, it is possible to construct a function xQ(ξ)and then express the three contributions (scalar, tensor and cosmic strings) to theCMB temperature anisotropies only as a function of the Fayet-Iliopoulos term ξ. Thus,we obtain that the total quadrupole temperature anisotropy reads

(

δT

T

)tot

Q

∼ ξ

M2Pl

π2

90g2x−4

Q f−2(x2Q)

W(x2Q(g2ξ)(λ2M2

Pl))[

1 + W(x2Q(g2ξ)(λ2M2

Pl))]2

+

(

0.77g

8√

2π

)2

+

(

9

4

)21/2

, (89)


QN =60

QN =70

QN =50

1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

0

2

4

6

8

10

c

xQ

Figure 6. Iso-contours of NQ for NQ = 50, 60, 70 in the plan (xQ, c ≡(g2ξ)/(λ2M2

Pl)). This graph is obtained for g = 10−2.

where the only unknown is ξ, for given values of g and λ. We normalise the quadrupoleanisotropy to COBE and we calculate numerically ξ, and thus xQ using the functionxQ(ξ), as well as the various contributions as a function of the couplings λ and g.

Our results are listed below. Firstly, as previously, it is straightforward to seethat the tensor contribution, even for g = 1, is completely negligible. Second, theinflaton field SQ is of the order of 10MPl for the studied parameter space in λ andg whereas MD =

√ξ is still of the order of 2 × 1015 GeV. Concerning cosmic strings

contribution to the CMB, we can see from Fig. 7 that it is not constant: it dependsstrongly on the value of the gauge coupling g and the superpotential coupling λ. Forg >∼ 1, it is not possible that the D-term inflationary era lasts 60 e-foldings. Thus, amultiple stage inflation is necessary to solve the horizon problem. For g >∼ 2 × 10−2,the cosmic strings contribution is always greater than the WMAP limit, thus, it isruled out.

For high values of the superpotential coupling λ >∼ 10−3, or for small values of thegauge coupling g (namely g <∼ 10−4), our findings are very similar to the ones obtainedfor D-term inflation within the SUSY context. This is expected since in these limits,

W

(

g2ξ

λ2M2Pl

z

)

=|S|2M2

Pl

≪ 1 . (90)

On the other hand, SUGRA corrections cannot be neglected for small values of λ,in which case the cosmic strings contribution rises again. It is thus possible to specify,for a given value of g, the allowed window for the superpotential coupling λ. As westated earlier, our analysis shows that the allowed cosmic strings contribution to theCMB temperature anisotropies, as imposed by the WMAP measurements, leads tothe constraint

g <∼ 2 × 10−2 . (91)

Provided the above condition is satisfied, the CMB constraint sets an upper bound to


0.5 1 1.5 2 2.5 3 3.5 4"###Ξ H´1015 GeVL

0.51

510

50100

Strings contribution H%L

1.´10-8 1.´10-6 0.0001Λ

0.1

1

10

100Cosmic strings contribution H%L

g=10−3

g=10−2

g=10−1

g=2x10−2

−4g=10

Figure 7. On the left, cosmic strings contribution as a function of the mass

scale√

ξ. This holds for all studied values of g. On the right, cosmicstrings contribution to the CMB temperature anisotropies, as a function of thesuperpotential coupling λ, for different values of the gauge coupling g. Themaximal contribution alowed by WMAP is represented by a dotted line. Theplots are derived in the framework of SUGRA.

the allowed window for the superpotential coupling λ, namely

λ <∼ 3 × 10−5 , (92)

up to a confidence level of 99%. This upper limit on λ is the same as our previous limitfound in the framework of SUSY. Note that the upper bound we found for λ, Eq. (92),is in agreement with the one found in Ref.[44] in a slightly different model, where thegauge coupling g is related to the Yukawa coupling λ through the relation λ =

√2g.

Our constraint on λ, Eq. (92), is also in agreement with the finding λ <∼ O(10−4−10−5)of Ref. [46].

Supergravity corrections induce a lower bound for λ but for g <∼ 2 × 10−3 itmust be very small and it is therefore uninteresting, if one wants to avoid fine tuningof the parameters. However, this lower limit is interesting for g in the range of[2 × 10−3, 2 × 10−2]. As an example, for g = 10−2, the CMB constraint imposes

10−8 <∼ λ <∼ 3 × 10−5 . (93)

This allowed window for λ shrinks as g goes from g = 10−2 to g = 2 × 10−2. This


is the most important contribution from supergravity. This fine tuning can be lesssevere if one invokes the curvaton mechanism [49].

As in the case of F-term inflation, it is also possible to convert this constraintinto a constraint on the mass scale, which is given by the Fayet-Iliopoulos term ξ andwe get

√

ξ <∼ 2 × 1015 GeV . (94)

The above constraint on ξ is independent of the value of the gauge coupling g, providedEq. (91) is satisfied.

We would like to bring to the attention of the reader that in the above study wehave neglected the quantum gravitational effects, i.e., a contribution [50]

[∆V (|S|)]QG = C1d2V

dS2

V

M2Pl

+ C2V 2

M4Pl

(95)

(with C1, C2 numerical coefficients of the order of 1), to the effective potential, eventhough SQ ∼ O(10MPl). Our analysis is however still valid, since the effectivepotential given in Eq. (80) satisfies the conditions V (|S|) ≪ M4

Pl and m2S =

d2V/dS2 ≪ M2Pl, and thus the quantum gravitational corrections [∆V (|S|)]QG are

very small as compared to the effective potential V D−SUGRAeff [50].

4. Conclusions and discussion

In the context of SUSY GUTs, cosmic strings — occasionally accompanied byembedded strings — are the outcome of SSB schemes, compatible with high energyphysics and cosmology. As it was explicitely shown in Ref. [5], strings are genericallyformed at the end of a supersymmetric hybrid inflationary era, which can be eitherF-term or D-term type, as we follow the patterns of SSBs from grand unified gaugegroups GGUT down to the standard model gauge group GSM × Z2. One should keepin mind, that there are mechanisms to avoid cosmic strings formation, which are notconsidered here since we focus on the most standard GUT and hybrid inflation models.The strings forming at the end of inflation have a mass which is proportional to theinflationary scale. However, current CMB temperature anisotropies data limit thecontribution of cosmic strings to the angular power spectrum. The aim of this studyis to use the data given by the realm of cosmology to constrain the parameters ofsupersymmetric hybrid inflationary models.

More precisely, in the framework of generic SUSY GUTs, we study theconsequences of simple hybrid inflation models in the context of some realisticcosmological scenario. We then compare the observational predictions of these modelsin order to constrain the parameter’s space. We perform our calculations within SUSYor SUGRA, whenever this is needed.

For F-term inflation, the symmetry breaking scale M associated with the inflatonmass scale (and the strings scale) is a slowly varying function of the superpotentialcoupling κ, and of the dimensionality N of the representations to which the scalarcomponents of the chiral Higgs superfields belong. For SO(10), under the requirementthat R-parity is conserved down to low energies, N = 126. For E6, the Higgsrepresentations can be N = 27 or N = 351. The dependence of a measurablequantity (the cosmic strings contribution) on this discrete parameter N can providean interesting tool to discriminate among different grand unified gauge groups.


The unique free parameter of the model, the dimensionless coupling κ, shouldobey two constraints. Namely, it should satisfy the gravitino constraint, and it shouldalso obey to the CMB constraint. We found that the CMB constraint on the coupling κis the strongest one: it imposes κ <∼ 7×10−7×(126/N ), for N ∈ 1,16,27,126,351whereas the gravitino constraint reads κ <∼ 8 × 10−3.

The WMAP constraint on the superpotential coupling κ can be expressed into aconstraint on the mass scale, namely M <∼ 2×1015 GeV. The value of the inflaton fieldis of the same order of magnitude, and since it is below the Planck scale, it impliesthat global supersymmetry is sufficient for our analysis. Thus, it is not necessary totake into account supergravity corrections.

The implications of this finding are quite strong. Supersymmetric hybridinflation was advertised to circumvent the naturalness issue appearing in the mostelegant nonsupersymmetric inflationary model, i.e., chaotic inflation. Namely, chaoticinflation needed a very small coupling (λ ∼ 10−14). Cosmology taught us thatsupersymmetric hybrid inflation also suffers from the fine tuning issue when embeddedwithin the class of cosmological models considered here.

In this study we also employed a mechanism where this problem can be lifted.This is done if we use the curvaton mechanism. In this case another scalar field, calledthe curvaton, could generate the initial density perturbations whereas the inflatonfield is only responsible for the dynamics of the Universe. Within supersymmetrictheories such scalar fields are expected to exist. In this case, the coupling κ is onlyconstrained by the gravitino limit.

We performed the same analysis for D-term inflation. In this case we foundthat the value of the inflaton field is of the order of the Planck scale or higher.One should therefore consider local supersymmetry, taking into account all the one-loop radiative corrections. To our knowledge, this was never considered in previousstudies. Our analysis gives a nonconstant contribution of cosmic strings to the CMBtemperature anisotropies, which is strongly dependent on the gauge coupling g andthe superpotential λ. We claim that the D-term inflationary model is still an openpossibility, since it does not always imply that the cosmic strings contribution tothe CMB is above the upper limit allowed by WMAP. To avoid contradiction withthe data, the free parameters of the model are strongly constrained. More precisely,we found that the gauge coupling must satisfy the constraint g <∼ 2 × 10−2, and thesuperpotential coupling must obey the condition λ <∼ 3 × 10−5. This is the samelimit as the one found in the SUSY framework. The supergravity corrections alsogive a lower limit on the value of this parameter. As an example, for g = 10−2 wefound 10−8 <∼ λ <∼ 10−8. This allowed window for λ shrinks as g goes from g = 10−2

to g = 2 × 10−2. The conditions imposed by the CMB data on the couplingsλ, g can be expressed as a single constraint on the Fayet-Iliopoulos term ξ, namely√ξ <∼ 2 × 1015 GeV, which remains valid independently of g.

To conclude, we would like to emphasise that cosmic strings of the GUT scaleare in agreement with observationnal data and can play a role in cosmology. CMBdata do not rule out cosmic strings; they impose strong constraints on their possiblecontribution. One should use these constraints to test high energy physics such assupersymmetric grand unified theories. This is indeed the philosophy of our paper.


Acknowledgments

It is a pleasure to thank P. Brax, N. Chatillon, G. Esposito-Farese, J. Garcia-Bellido,R. Jeannerot, A. Linde, J. Martin, P. Peter, and C. -M. Viallet for discussions andcomments.

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http://arXiv.org/abs/hep-ph/0111328


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(1990), Hardwood Academic Publishers.

5. Appendix

We list below the SSB schemes compatible with high energy physics and cosmology, asgiven in detail in Ref. [5]. Let us remind to the reader that every

n−→ represent an SSBduring which there is formation of topological defects. Their nature is given by n: 1for monopoles, 2 for topological cosmic strings, 2′ for embedded strings, 3 for domainwalls. Please note also that for e.g. 3C 2L 2R 1B−L stands for the group SU(3)C×SU(2)L× SU(2) R× U(1)B−L. We refer the reader to Ref. [5] for more details.

We first give the SSB schemes of SO(10) down to the GSM × Z2.

SO(10)

1−→ 5 1V1−→ 3C 2L 1Z 1V

2−→ GSM Z21−→ 4C 2L 2R −→ Eq. (97)

1,2−→ 4C 2L 2R ZC2 −→ Eq. (98)

1,2−→ 4C 2L 1R ZC2 −→ · · ·

1−→ 4C 2L 1R −→ · · ·1,2−→ 3C 2L 2R 1B−L Z

C2 −→ · · ·

1−→ 3C 2L 2R 1B−L −→ · · ·1−→ 3C 2L 1R 1B−L

2−→ GSM Z2

(96)

where

4C 2L 2R

1−→ 3C 2L 2R 1B−L

1−→ 3C 2L 1R 1B−L2−→ GSM Z2

2′,2−→ GSM Z2

1−→ 4C 2L 1R

1−→ 3C 2L 1R 1B−L2−→ GSM Z2

2′,2−→ GSM Z21−→ 3C 2L 1R 1B−L

2−→ GSM Z2

(97)

and

4C 2L 2R ZC2

1−→ 3C 2L 2R 1B−L ZC2

3−→ 3C 2L 2R 1B−L −→ · · ·1,3−→ 3C 2L 1R 1B−L

2−→ GSM Z2

1−→ 4C 2L 1R ZC2

3−→ 4C 2L 1R −→ · · ·1,3−→ 3C 2L 1R 1B−L

2−→ GSM Z23−→ 4C 2L 2R −→ Eq. (97)1−→ 4C 2L 1R −→ · · ·

1,3−→ 3C 2L 2R 1B−L −→ · · ·1,3−→ 3C 2L 1R 1B−L

2−→ GSM Z2

(98)


We then proceed with the list of the SSB schemes of E6 down to the GSM × Z2.We first give the SSB patterns where E6 is broken via SO(10) × U(1).

E61→ 10 1V′

2−→ 10 −→ · · ·1−→ 5 1V 1V′ −→ Eq. (101)1−→ 5F 1V 1V′ −→ Eq. (102)1−→ 5E 1V 1V′

2′,2−→ GSM Z22−→ 5 1V′ Z2 −→ · · ·

1,2−→ 5 1V −→ · · ·1−→ 5F 1V

2′,2−→ GSM Z21−→ GSM 1V

2−→ GSM Z21,2−→ GSM 1V′ Z2

2−→ GSM Z21,2−→ 4C 2L 2R 1V′ −→ Eq. (103)1−→ 4C 2L 2R −→ Eq. (97)1−→ 3C 2L 2R 1B−L 1V′ −→ · · ·1−→ 3C 2L 1R 1B−L 1V′ −→ · · ·

(99)

with more direct schemes

E6

1−→ 5 1V 1V′ −→ Eq.(101)1−→ 5F 1V 1V′ −→ Eq.(102)1−→ 5E 1V 1V′

2′,2−→ GSM Z21−→ 5 1V −→ · · ·1−→ 5 1V′ −→ · · ·1−→ 5F 1V

2′,2−→ GSM Z21−→ 4C 2L 2R 1V′ −→ Eq. (103)1−→ 4C 2L 2R −→ Eq. (97)1−→ 4C 2L 1R −→ · · ·1−→ 4C 2L 1R 1V′ −→ · · ·1−→ 3C 2L 2R 1B−L 1V′ −→ · · ·1−→ 3C 2L 1R 1B−L 1V′ −→ · · ·1−→ 3C 2L 1R 1B−L

2−→ GSM Z21−→ GSM 1V

2−→ GSM Z21,2−→ GSM 1V′ Z2

2−→ GSM Z2

(100)

where

5 1V 1V′

2−→ 5 1V′ Z2

1−→ GSM 1V′ Z22−→ GSM Z2

1−→ GSM 1V 1V′

2−→ GSM 1V2−→ GSM Z2

2−→ GSM 1V′ Z22−→ GSM Z2

2−→ 5 1V −→ · · ·

(101)

5F 1V 1V′

2−→ 5F 1V2′,2−→ GSM Z2

2′,2−→ GSM Z2

(102)


4C 2L 2R 1V′

2−→ 4C 2L 2R −→ Eq. (97)

1−→ 3C 2L 1R 1B−L 1V′

2−→ 3C 2L 1R 1B−L2−→ GSM Z2

2′,2−→ GSM1V′ Z22−→ GSM Z2

2′,2−→ GSM Z2

1−→ 3C 2L 2R 1B−L 1V′

1−→ 3C 2L 1R 1B−L 1V′ −→ · · ·2−→ 3C 2L 2R 1B−L −→ · · ·

2′,2−→ GSM1V′ Z22−→ GSM Z2

1,2−→ 3C 2L 1R 1B−L2−→ GSM Z2

2′,2−→ GSM Z2

1−→ 4C 2L 1R 1V′

2−→ 4C 2L 1R −→ · · ·1−→ 3C 2L 1R 1B−L 1V′ −→ · · ·

2′,2−→ GSM1V′ Z22−→ GSM Z2

1,2−→ 3C 2L 1R 1B−L2−→ GSM Z2

2−→ GSM Z21,2−→ GSM1V′ Z2

2−→ GSM Z21,2−→ 3C 2L 2R 1B−L −→ · · ·1−→ 3C 2L 1R 1B−L

2−→ GSM Z2

(103)

We proceed with the allowed SSB patterns of E6 down to the GSM × Z2 viaSU(3)C × SU(3)L × SU(3)R.

E60→ 3C 3L 3R

1−→ 3C 2L 2(R) 1Y(R)1YL −→ Eq. (106)

1−→ 3C 3L 2(R)1(YR) −→ Eq. (107)1−→ 3C 2L 3R 1YL −→ Eq. (108)1−→ 3C 3L 1(R) 1Y(R)

−→ Eq. (109)1−→ 3C 2L 1(R) 1Y(R)

1YL −→ · · ·1−→ 3C 2L 2(R) 1B−L −→ · · ·1−→ 3C 2L 1(R) 1B−L

2−→ GSM Z2

(104)

with more direct breakings

E6

1−→ 3C 2L 2(R) 1YL 1Y(R)−→ Eq. (106)

1−→ 3C 3L 2(R) 1Y(R)−→ Eq. (107)

1−→ 3C 2L 3R 1YL −→ Eq. (108)1−→ 3C 2L 1(R) 1YL 1Y(R)

−→ · · ·1−→ 3C 2L 2(R) 1B−L −→ · · ·1−→ 3C 2L 1(R) 1B−L

2−→ GSM Z2

(105)


where

3C 2L 2(R) 1YL 1Y(R)

1−→ 3C 2L 2(R) 1B−L

1−→ 3C 2L 1(R) 1B−L2−→ GSM Z2

2′,2−→ GSM Z2

1−→ 3C 2L 1(R) 1Y(R)1YL

2−→ 3C 2L 1(R) 1B−L2−→ GSM Z2

2−→ GSM Z21,2−→ 3C 2L 1(R) 1B−L

2−→ GSM Z2

2′,2−→ GSM Z2

(106)

3C 3L 2(R)1(YR)

1−→ 3C 2L 2(R) 1Y(R)1YL −→ Eq. (106)

1−→ 3C 2L 1(R) 1Y(R)1YL −→ · · ·

1−→ 3C 2L 2(R) 1B−L −→ · · ·1−→ 3C 2L 1(R) 1B−L

2−→ GSM Z2

2′,2−→ GSM Z2

(107)

3C 2L 3R 1YL

1−→ 3C 2(R) 2L 1YL 1Y(R)−→ Eq. (106)

2′

−→ 3C 2(R) 2L 1B−L −→ · · ·1−→ 3C 2L 1(R) 1YL 1Y(R)

−→ · · ·1−→ 3C 2L 1(R) 1B−L

2−→ GSM Z2

2′,2−→ GSM Z2

(108)

and

3C 3L 1(R) 1Y(R)

1−→ 3C 2L 1YL 1(R) 1Y(R)−→ · · ·

2′

−→ 3C 2L 1(R) 1B−L2−→ GSM Z2

2′,2−→ GSM Z2

(109)

Finally, we list below the SSB schemes of E6 down to the GSM×Z2 via SU(6)×SU(2).These are:

E60−→ 6 2L

or E6

1−→ 3C 3R 2L 1YL −→ Eq. (108)1−→ 4C 2L 2R 1V′ −→ Eq. (103)0−→ 4C 2L 2R −→ · · ·1−→ 4C 2L 1R 1V′ −→ · · ·1−→ 4C 2L 1R −→ · · ·1−→ 3C 2L 2(R) 1B−L −→ · · ·1−→ 3C 2L 1(R) 1YL 1Y(R)

−→ · · ·1−→ 3C 2L 1(R) 1B−L

2−→ GSM Z2

(110)

and

E60−→ 6 2R

or E6

1−→ 4C 2L 2R 1V ′ −→ Eq. (103)1−→ 4C 2L 1R 1V′ −→ · · ·0−→ 4C 2L 2R −→ · · ·1−→ 4C 2L 1R −→ · · ·1−→ 3C 2L 2R 1B−L 1V′ −→ · · ·1−→ 3C 2L 2R 1B−L −→ · · ·1−→ 3C 2L 1R 1B−L

2−→ GSM Z2

(111)


As it was shown in Ref. [5], the SSB schemes of SU(6) and SU(7) down tothe standard model which could accommodate an inflationary era with no defect(of any kind) at later times are inconsistent with proton lifetime measurements andminimal SU(6) and SU(7) does not predict neutrino masses. Thus, these models areincompatible with high energy physics phenomenology.

Constraints on supersymmetric grand unified theories from cosmology

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